We investigate the existence of the periodic solutions of a nonlinear integro-differential system with piecewise alternately advanced and retarded argument of generalized type, in short DEPCAG; that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. In the process we use Green’s function for periodic solutions and convert the given DEPCAG into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii’s fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. We also use the contraction mapping principle to show the existence of a unique periodic solution. Appropriate examples are given to show the feasibility of our results.

Among the functional differential equations, Myshkis [

Let

Let

In 2008, Akhmet et al. [

Recently, Chiu and Pinto [

It is well-known that there are many subjects in physics and technology using mathematical methods that depend on the linear and nonlinear integro-differential equations, and it became clear that the existence of the periodic solutions and its algorithm structure from more important problems in the present time. Where many of studies and researches [

Samoilenko and Ronto [

Butris [

In the current paper, we study the existence of periodic solutions of a nonlinear integro-differential system with piecewise alternately advanced and retarded argument:

In our paper we assume that the solutions of the nonlinear integro-differential systems with DEPCAG (

The rest of the paper is organized as follows. In Section

In this section we state and define Green's function for periodic solutions of the nonlinear integro-differential system with piecewise alternately advanced and retarded argument (

Let

For every

From now on the following assumption will be needed.

The homogenous equation

For

Now, we solve the DEPCAG system (

Suppose that the condition (

We note that the condition (

To prove double

Suppose that the condition (

For Lemma

Suppose that the condition (

In this section, we prove the main theorems of this paper, so we recall the nonlinear integro-differential systems with DEPCAG (

Consider

Let

where

There exists a continuous function

Moreover,

For

Moreover,

For

Moreover,

For every

where

For every

where

For every

There exists

there exists

Note that

For any

For

Then,

Using Definition

Suppose that the conditions (

Consider the operator

It is easy to see that the DEPCAG system (

To prove some existence criteria for

Next we state first Krasnoselskii's fixed point theorem which enables us to prove the existence of a periodic solution. For the proof of Krasnoselskii's fixed point theorem we refer the reader to [

Let

Then there exists

Krasnoselskii’s theorem may be combined with Banach and Schauder's fixed point theorems. In a certain sense, we can interpret this as follows: if a compact operator has the fixed point property, under a small perturbation, then this property can be inherited. The theorem is useful in establishing the existence results for perturbed operator equations. It also has a wide range of applications to nonlinear integral equations of mixed type for proving the existence of solutions. Thus the existence of fixed points for the sum of two operators has attracted tremendous interest, and their applications are frequent in nonlinear analysis. See [

We note that to apply Krasnoselskii's fixed point theorem we need to construct two mappings; one is contraction and the other is compact. Therefore, we express (

If (

Let

Let

Similarly,

If (

If (

As the operator

The function

Let

In a similar way, for

If (

Suppose the hypotheses (

By Lemma

Next, we prove that if

Let

We now see that all the conditions of Krasnoselskii's theorem are satisfied. Thus there exists a fixed point

By the symmetry of the conditions, we will obtain as Theorem

Suppose the hypothesis (

By Lemma

Next, we prove that if

Let

Applying Banach's fixed point theorem we have the following.

Suppose the hypotheses (

Let the mapping

This completes the proof by invoking the contraction mapping principle.

As a direct consequence of the method, Schauder's theorem implies the following.

Suppose the hypotheses (

Considering the DEPCAG system (

Considering the nonlinear system of differential equations with a general piecewise alternately advanced and retarded argument,

Suppose that (

To determine criteria for the existence and uniqueness of subharmonic solutions of the DEPCAG system (

There exists

There exists

As immediate corollaries of Theorems

Suppose the hypotheses (

Suppose the hypotheses (

Suppose the hypotheses (

Suppose the hypotheses (

We will introduce appropriate examples in this section. These examples will show the feasibility of our theory.

Mathematical modelling of real-life problems usually results in functional equations, like ordinary or partial differential equations, integral and integro-differential equations, and stochastic equations. Many mathematical formulations of physical phenomena contain integro-differential equations; these equations arise in many fields like fluid dynamics, biological models, and chemical kinetics. So, we first consider nonlinear integro-differential equations with a general piecewise constant argument mentioned in the introduction and obtain some new sufficient conditions for the existence of the periodic solutions of these systems.

Let

The conditions of Theorem

Indeed, for any

Then, by Theorem

Thus many examples can be constructed where our results can be applied.

Let

The hypotheses of Theorem

Note that similar results can be obtained under (

Let us consider another example for second-order differential equations with a general piecewise constant argument. In this case, we can show the existence and uniqueness of periodic solutions of the following nonlinear DEPCAG system.

Consider the following nonlinear DEPCAG system:

We write the DEPCAG system (

The author declares that there is no conflict of interests regarding the publication of this paper.

This research was in part supported by FIBE 01-12 DIUMCE.